Verification and Validation in Scientific Computing

"Advances in scientific computing have made modelling and simulation an important part of the decision-making process in engineering, science, and public policy. This book provides a comprehensive and systematic development of the basic concepts, principles, and procedures for verification and validation of models and simulations. The emphasis is placed on models that are described by partial differential and integral equations and the simulations that result from their numerical solution. The methods described can be applied to a wide range of technical fields, from the physical sciences, engineering and technology and industry, through to environmental regulations and safety, product and plant safety, financial investing, and governmental regulations. This book will be genuinely welcomed by researchers, practitioners, and decision makers in a broad range of fields, who seek to improve the credibility and reliability of simulation results. It will also be appropriate either for university courses or for independent study"--Résumé de l'éditeur Cover Half-title Title Copyright Dedication Contents Preface Acknowledgments 1 Introduction 1.1 Historical and modern role of modeling and simulation 1.1.1 Historical role of modeling and simulation 1.1.2 Changing role of scientific computing in engineering 1.1.2.1 Changing role of scientific computing in design, performance and safety of engineering systems 1.1.2.2 Interaction of scientific computing and experimental investigations 1.1.3 Changing role of scientific computing in various fields of science 1.2 Credibility of scientific computing 1.2.1 Computer speed and maturity of scientific computing 1.2.2 Perspectives on credibility of scientific computing 1.2.3 How is credibility built in scientific computing? 1.2.3.1 Quality of the analysts conducting the scientific computing 1.2.3.2 Quality of the physics modeling 1.2.3.3 Verification and validation activities 1.2.3.4 Uncertainty quantification and sensitivity analyses 1.3 Outline and use of the book 1.3.1 Structure of the book 1.3.2 Use of the book in undergraduate and graduate courses 1.3.3 Use of the book by professionals 1.4 References Part I Fundamental concepts 2 Fundamental concepts and terminology 2.1 Development of concepts and terminology 2.1.1 Early efforts of the operations research community 2.1.2 IEEE and related communities 2.1.3 US Department of Defense community 2.1.4 AIAA and ASME communities 2.1.4.1 AIAA Guide 2.1.4.2 ASME Guide 2.1.5 Hydrology community 2.2 Primary terms and concepts 2.2.1 Code verification 2.2.2 Solution verification 2.2.3 Model validation 2.2.4 Predictive capability 2.2.5 Calibration 2.2.6 Certification and accreditation 2.3 Types and sources of uncertainties 2.3.1 Aleatory uncertainty 2.3.2 Epistemic uncertainty 2.3.2.1 Recognized uncertainty 2.3.2.2 Blind Uncertainty 2.4 Error in a quantity 2.5 Integration of verification, validation, and prediction 2.5.1 Specification of the application of interest 2.5.2 Planning and prioritization of activities 2.5.3 Code verification and software quality assurance activities 2.5.4 Design and execution of validation experiments 2.5.5 Computation of the system response quantities and solution verification 2.5.6 Computation of validation metric results 2.5.7 Prediction and uncertainty estimation for the application of interest 2.5.8 Assessment of model adequacy 2.5.9 Documentation of M&S activities 2.6 References 3 Modeling and computational simulation 3.1 Fundamentals of system specifications 3.1.1 Systems and surroundings Example 1: Orbiting spacecraft Example 2: Beam deflection Example 3: Electronic circuit 3.1.2 Environments and scenarios 3.2 Fundamentals of models and simulations 3.2.1 Goals of scientific computing 3.2.2 Models and simulations 3.2.3 Importance of nondeterministic simulations 3.2.4 Analysis of nondeterministic systems 3.2.5 Example problem: mechanical oscillation 3.2.5.1 Aleatory uncertainty 3.2.5.2 Epistemic uncertainty 3.3 Risk and failure 3.4 Phases of computational simulation 3.4.1 Conceptual modeling phase 3.4.2 Mathematical modeling phase 3.4.3 Discretization and algorithm selection phase 3.4.4 Computer programming phase 3.4.5 Numerical solution phase 3.4.6 Solution representation phase 3.5 Example problem: missile flight dynamics 3.5.1 Conceptual modeling phase 3.5.2 Mathematical modeling phase 3.5.3 Discretization and algorithm selection phase 3.5.4 Computer programming phase 3.5.5 Numerical solution phase 3.5.6 Solution representation phase 3.6 References Part II Code verification 4 Software engineering 4.1 Software development 4.1.1 Software process models 4.1.2 Architectural design 4.1.3 Programming languages 4.1.4 Agile programming 4.1.5 Software reuse 4.1.6 Refactoring 4.2 Version control 4.3 Software verification and validation 4.3.1 Definitions 4.3.2 Static analysis 4.3.2.1 Software inspection 4.3.2.2 Compiling the code 4.3.2.3 Automatic static analyzers 4.3.3 Dynamic testing 4.3.3.1 Defect testing Unit testing Component testing System testing 4.3.3.2 Regression testing 4.3.3.3 Software validation testing 4.3.4 Test harness and test suites 4.3.5 Code coverage 4.3.6 Formal methods 4.4 Software quality and reliability 4.4.1 Reliability metrics 4.4.1.1 Defect density analysis 4.4.1.2 Complexity analysis Lines of source code NPATH metric Cyclomatic complexity Depth of conditional nesting Depth of inheritance tree 4.5 Case study in reliability: the T experiments 4.6 Software engineering for large software projects 4.6.1 Software requirements 4.6.1.1 Types of software requirements 4.6.1.2 Requirements engineering process 4.6.1.3 Requirements management 4.6.2 Software management 4.6.2.1 Project management 4.6.2.2 Cost estimation 4.6.2.3 Configuration management 4.6.2.4 Quality management 4.6.2.5 Process improvement 4.7 References 5 Code verification 5.1 Code verification criteria 5.1.1 Simple tests 5.1.1.1 Symmetry tests 5.1.1.2 Conservation tests 5.1.1.3 Galilean invariance tests 5.1.2 Code-to-code comparisons 5.1.3 Discretization error evaluation 5.1.4 Convergence tests 5.1.5 Order-of-accuracy tests 5.2 Definitions 5.2.1 Truncation error 5.2.1.1 Example: truncation error analysis 5.2.1.2 Generalized truncation error expression (GTEE) 5.2.2 Discretization error 5.2.3 Consistency 5.2.4 Stability 5.2.5 Convergence 5.3 Order of accuracy 5.3.1 Formal order of accuracy 5.3.2 Observed order of accuracy 5.3.2.1 Asymptotic range 5.3.2.2 Effects of iterative and round-off error 5.4 Systematic mesh refinement 5.4.1 Uniform mesh refinement 5.4.2 Consistent mesh refinement 5.4.3 Mesh transformations 5.4.4 Mesh topology issues 5.5 Order verification procedures 5.5.1 Spatial discretization 1 Define mathematical model 2 Choose numerical algorithm 3 Establish formal order of accuracy 4 Obtain exact solution to mathematical model 5 Obtain numerical solutions on at least four meshes 6 Compute observed order of accuracy 7 Fix test implementation 8 Debug the code 9 Document results 5.5.2 Temporal discretization 5.5.3 Spatial and temporal discretization 5.5.3.1 Separate order analysis 5.5.3.2 Combined order analysis 5.5.4 Recommendations for debugging 5.5.5 Limitations of order verification 5.5.6 Alternative approaches for order verification 5.5.6.1 Residual method 5.5.6.2 Statistical method 5.5.6.3 Downscaling method 5.5.6.4 Summary of order verification approaches 5.6 Responsibility for code verification 5.7 References 6 Exact solutions 6.1 Introduction to differential equations 6.2 Traditional exact solutions 6.2.1 Procedures 6.2.1.1 Separation of variables 6.2.1.2 Transformations 6.2.1.3 Method of characteristics 6.2.1.4 Advanced approaches 6.2.2 Example exact solution: 1-D unsteady heat conduction 6.2.3 Example with order verification: steady Burgers’ equation 6.2.4 Example with order verification: linear elasticity 6.3 Method of manufactured solutions (MMS) 6.3.1 Procedure 6.3.1.1 Manufactured solution guidelines for code verification 6.3.1.2 Boundary and initial conditions 6.3.2 Benefits of MMS for code verification 6.3.3 Limitations of MMS for code verification 6.3.4 Examples of MMS with order verification 6.3.4.1 2-D steady heat conduction 6.3.4.2 2D Steady Euler equations 6.4 Physically realistic manufactured solutions 6.4.1 Theory-based solutions 6.4.2 Method of nearby problems (MNP) 6.4.2.1 Procedure 6.4.2.2 Example exact solution: 2-D steady Navier–Stokes equations 6.5 Approximate solution methods 6.5.1 Infinite series solutions 6.5.2 Reduction to ordinary differential equations 6.5.3 Benchmark numerical solutions 6.5.4 Example series solution: 2-D steady heat conduction 6.5.5 Example benchmark convergence test: 2-D hypersonic flow 6.6 References Part III Solution verification 7 Solution verification 7.1 Elements of solution verification 7.2 Round-off error 7.2.1 Floating point representation 7.2.2 Specifying precision in a code 7.2.2.1 C/C++ programming languages 7.2.2.2 Fortran 95/2003 programming languages 7.2.2.3 MATLAB® Programming Language 7.2.3 Practical guidelines for estimating round-off error 7.3 Statistical sampling error 7.3.1 Estimation of statistical sampling error 7.4 Iterative error 7.4.1 Iterative methods 7.4.1.1 Equations with a single unknown 7.4.1.2 Systems of equations Direct solution methods Stationary iterative methods Krylov subspace methods Hybrid methods Examples of iterative methods in scientific computing 7.4.2 Iterative convergence 7.4.2.1 Types of iterative convergence Monotone convergence Oscillatory convergence General convergence 7.4.2.2 Iterative convergence criteria Difference between iterates Iterative residuals 7.4.3 Iterative error estimation 7.4.3.1 Machine zero method 7.4.3.2 Local convergence rate Monotone iterative convergence Oscillatory iterative convergence 7.4.4 Relation between iterative residuals and iterative error 7.4.5 Practical approach for estimating iterative error 7.5 Numerical error versus numerical uncertainty 7.6 References 8 Discretization error 8.1 Elements of the discretization process 8.1.1 Discretization of the mathematical model 8.1.1.1 The finite difference method 8.1.1.2 The finite volume method 8.1.1.3 The finite element method 8.1.2 Discretization of the domain 8.1.2.1 Structured meshes 8.1.2.2 Unstructured meshes 8.1.2.3 Cartesian meshes 8.1.2.4 Mesh-free methods 8.2 Approaches for estimating discretization error 8.2.1 Type I: Higher-order estimates 8.2.1.1 Mesh refinement methods 8.2.1.2 Order refinement methods 8.2.1.3 Finite element recovery methods 8.2.2 Type II: Residual-based methods 8.2.2.1 Error transport equations Continuous discretization error transport equation Discrete discretization error transport equation Approximating the truncation error System response quantities 8.2.2.2 Finite element residual methods Explicit residual methods Implicit residual methods 8.2.2.3 Adjoint methods for system response quantities Adjoint methods in the finite element method Adjoint methods in the finite volume method 8.3 Richardson extrapolation 8.3.1 Standard Richardson extrapolation 8.3.2 Generalized Richardson extrapolation 8.3.3 Assumptions 8.3.3.1 Asymptotic range 8.3.3.2 Uniform mesh spacing 8.3.3.3 Systematic mesh refinement 8.3.3.4 Smooth solutions 8.3.3.5 Other numerical errors sources 8.3.4 Extensions 8.3.4.1 Completed Richardson extrapolation in space 8.3.4.2 Completed Richardson extrapolation in space and time 8.3.4.3 Least squares extrapolation 8.3.5 Discretization error estimation 8.3.5.1 Example: Richardson extrapolation-based error estimation 8.3.6 Advantages and disadvantages 8.4 Reliability of discretization error estimators 8.4.1 Asymptotic range 8.4.2 Observed order of accuracy 8.4.2.1 Constant grid refinement factor 8.4.2.2 Non-constant grid refinement factor 8.4.2.3 Application to system response quantities 8.4.2.4 Application to local quantities 8.5 Discretization error and uncertainty 8.6 Roache’s grid convergence index (GCI) 8.6.1 Definition 8.6.2 Implementation 8.6.3 Variants of the GCI 8.6.3.1 Least squares method 8.6.3.2 Global averaging method 8.6.3.3 Factor of safety method 8.6.4 Reliability of the GCI 8.7 Mesh refinement issues 8.7.1 Measuring systematic mesh refinement 8.7.2 Grid refinement factor 8.7.3 Fractional uniform refinement 8.7.4 Refinement vs. coarsening 8.7.5 Unidirectional refinement 8.8 Open research issues 8.8.1 Singularities and discontinuities 8.8.2 Oscillatory convergence with mesh refinement 8.8.3 Multi-scale models 8.8.4 Coarse grid error estimators 8.9 References 9 Solution adaptation 9.1 Factors affecting the discretization error 9.1.1 Relating discretization error to truncation error 9.1.2 1-D truncation error analysis on uniform meshes 9.1.3 1-D truncation error analysis on nonuniform meshes 9.1.4 Isotropic versus anisotropic mesh adaptation 9.2 Adaptation criteria 9.2.1 Solution features 9.2.2 Discretization error 9.2.3 Recovery methods 9.2.4 Truncation errorresiduals 9.2.4.1 General truncation errorresidual-based methods 9.2.4.2 Finite element residual-based methods 9.2.5 Adjoint-based adaptation 9.3 Adaptation approaches 9.3.1 Adaptive remeshing 9.3.2 Mesh adaptation 9.3.2.1 Local mesh refinementcoarsening (h-adaptation) 9.3.2.2 Mesh movement (r-adaptation) 9.3.2.3 Mixed mesh refinement (r- and h-adaptation) 9.3.3 Order refinement (p-adaptation) 9.4 Comparison of methods for driving mesh adaptation 9.4.1 Mathematical model 9.4.2 Exact solution 9.4.3 Discretization approach 9.4.4 Results 9.5 References Part IV Model validation and prediction 10 Model validation fundamentals 10.1 Philosophy of validation experiments 10.1.1 Validation experiments vs. traditional experiments 10.1.2 Goals and strategy of validation 10.1.2.1 Scientific validation 10.1.2.2 Project-oriented validation 10.1.3 Sources of error in experiments and simulations 10.1.4 Validation using data from traditional experiments 10.2 Validation experiment hierarchy 10.2.1 Characteristics of the complete system tier 10.2.2 Characteristics of the subsystem tier 10.2.3 Characteristics of the benchmark tier 10.2.4 Characteristics of the unit problem tier 10.2.5 Construction of a validation hierarchy 10.3 Example problem: hypersonic cruise missile 10.3.1 System tier 10.3.2 Subsystem tier 10.3.3 Benchmark tier 10.3.4 Unit-problem tier 10.3.5 Validation pyramid 10.3.6 Final comments 10.4 Conceptual, technical, and practical difficulties of validation 10.4.1 Conceptual difficulties 10.4.2 Technical and practical difficulties 10.5 References 11 Design and execution of validation experiments 11.1 Guidelines for validation experiments 11.1.1 Joint effort between analysts and experimentalists 11.1.2 Measurement of all needed input data 11.1.3 Synergism between computation and experiment 11.1.4 Independence and dependence between computation and experiment 11.1.5 Hierarchy of experimental measurements 11.1.6 Estimation of experimental uncertainty 11.2 Validation experiment example: Joint Computational/Experimental Aerodynamics Program (JCEAP) 11.2.1 Basic goals and description of JCEAP 11.2.2 Joint planning and design of the experiment 11.2.2.1 Wind tunnel conditions 11.2.2.2 Model geometry 11.2.2.3 Model fabrication and instrumentation 11.2.3 Characterize boundary conditions and system data 11.2.4 Synergism between computation and experiment 11.2.5 Independence and dependence between computation and experiment 11.2.6 Hierarchy of experimental measurements 11.3 Example of estimation of experimental measurement uncertainties in JCEAP 11.3.1 Random and systematic uncertainties 11.3.2 Example of DOE procedure for JCEAP force and moment experiment 11.3.2.1 DOE principles 11.3.2.2 DOE analysis and results 11.3.3 Example of DOE procedure for JCEAP surface pressure experiment 11.3.3.1 DOE principles 11.3.3.2 DOE analysis and results 11.4 Example of further computational–experimental synergism in JCEAP 11.4.1 Assessment of computational submodels 11.4.1.1 Transport property submodels 11.4.1.2 Equation of state submodel 11.4.1.3 Thermodynamic submodel 11.4.1.4 Continuum flow assumption 11.4.1.5 Outflow boundary condition assumption 11.4.1.6 Axisymmetric flow assumption 11.4.1.7 Re-evaluation of the experimental data 11.4.2 Simulation of the flowfield nonuniformities 11.4.2.1 Use of the flowfield calibration data 11.4.2.2 Simulation using the nonuniform flowfield 11.4.3 Lessons learned for validation experiments 11.5 References 12 Model accuracy assessment 12.1 Elements of model accuracy assessment 12.1.1 Methods of comparing simulations and experiments 12.1.2 Uncertainty and error in model accuracy assessment 12.1.3 Relationship between model accuracy assessment, calibration, and prediction 12.2 Approaches to parameter estimation and validation metrics 12.2.1 Parameter estimation 12.2.2 Hypothesis testing 12.2.3 Bayesian updating 12.2.4 Comparison of mean values 12.2.5 Comparison of probability distributions and p-boxes 12.3 Recommended features for validation metrics 12.3.1 Influence of numerical solution error 12.3.2 Assessment of the physics-modeling assumptions 12.3.3 Inclusion of experimental data post-processing 12.3.4 Inclusion of experimental uncertainty estimation 12.3.5 Inclusion of aleatory and epistemic uncertainties 12.3.6 Exclusion of any type of adequacy implication 12.3.7 Properties of a mathematical metric 12.4 Introduction to the approach for comparing means 12.4.1 Perspectives of the present approach 12.4.2 Development of the fundamental equations 12.4.3 Construction of the validation metric for one condition 12.4.4 Example problem: thermal decomposition of foam 12.5 Comparison of means using interpolation of experimental data 12.5.1 Construction of the validation metric over the range of the data 12.5.2 Global metrics 12.5.3 Example problem: turbulent buoyant plume 12.6 Comparison of means requiring linear regression of the experimental data 12.6.1 Construction of the validation metric over the range of the data 12.6.2 Global metrics 12.6.3 Example problem: thermal decomposition of foam 12.7 Comparison of means requiring nonlinear regression of the experimental data 12.7.1 Construction of the nonlinear regression equation 12.7.2 Computation of simultaneous confidence intervals for the metric 12.7.3 Global metrics 12.7.4 Example problem: compressible turbulent mixing 12.7.4.1 Problem description 12.7.4.2 Experimental data 12.7.4.3 Mathematical model 12.7.4.4 Validation metric results 12.7.5 Observations on the present approach 12.8 Validation metric for comparing p-boxes 12.8.1 Traditional methods for comparing distributions 12.8.2 Method for comparing p-boxes 12.8.2.1 Discussion of p-boxes 12.8.2.2 Validation metric for p-boxes 12.8.3 Pooling incomparable CDFs 12.8.3.1 u-pooling 12.8.3.2 Statistical significance of a metric 12.8.4 Inconsistency between experimental and simulation CDFs 12.8.5 Dealing with epistemic uncertainty in the comparisons 12.8.5.1 Epistemic uncertainty in the prediction and measurements 12.8.5.2 Epistemic and aleatory uncertainty in the metric 12.9 References 13 Predictive capability 13.1 Step 1: identify all relevant sources of uncertainty 13.1.1 Model inputs 13.1.2 Model uncertainty 13.1.3 Example problem: heat transfer through a plate 13.1.4 Final comments on step 1 13.2 Step 2: characterize each source of uncertainty 13.2.1 Model input uncertainty 13.2.2 Model uncertainty 13.2.3 Example problem: heat transfer through a solid 13.2.3.1 Model input uncertainty 13.2.3.2 Model uncertainty 13.3 Step 3: estimate numerical solution error 13.3.1 Iterative error 13.3.1.1 Iterative methods 13.3.1.2 Practical difficulties 13.3.2 Discretization error 13.3.2.1 Temporal discretization error 13.3.2.2 Finite-element-based methods for mesh convergence 13.3.2.3 Richardson extrapolation error estimators for mesh convergence 13.3.2.4 Practical difficulties 13.3.3 Estimate of total numerical solution error 13.3.4 Example problem: heat transfer through a solid 13.3.4.1 Iterative and discretization error estimation 13.3.4.2 Iterative and discretization error results 13.4 Step 4: estimate output uncertainty 13.4.1 Monte Carlo sampling of input uncertainties 13.4.1.1 Monte Carlo sampling for aleatory uncertainties 13.4.1.2 Monte Carlo sampling for combined aleatory and epistemic uncertainties 13.4.2 Combination of input, model, and numerical uncertainty 13.4.2.1 Combination of input and model uncertainty 13.4.2.2 Estimation of model uncertainty using alternative plausible models 13.4.2.3 Inclusion of numerical solution uncertainty 13.4.3 Example problem: heat transfer through a solid 13.4.3.1 Input uncertainties 13.4.3.2 Combination of input, model, and numerical uncertainties 13.5 Step 5: conduct model updating 13.5.1 Types of model parameter 13.5.2 Sources of new information 13.5.3 Approaches to parameter updating 13.5.4 Parameter updating, validation, and predictive uncertainty 13.5.4.1 Parameter updating 13.5.4.2 Validation after parameter updating 13.6 Step 6: conduct sensitivity analysis 13.6.1 Local sensitivity analysis 13.6.2 Global sensitivity analysis 13.7 Example problem: thermal heating of a safety component 13.7.1 Step 1: identify all relevant sources of uncertainty 13.7.2 Step 2: characterize each source of uncertainty 13.7.2.1 Model input uncertainty 13.7.2.2 Model uncertainty Possible temperature dependence of material properties Characterization of model uncertainty 13.7.3 Step 4: estimate output uncertainty 13.7.3.1 General discussion of combining input and model uncertainty 13.7.3.2 Combining input and model uncertainty for the thermal heating problem 13.7.3.3 Predicted probabilities for the regulatory condition 13.8 Bayesian approach as opposed to PBA 13.9 References Part V Planning, management, and implementation issues 14 Planning and prioritization in modeling and simulation 14.1 Methodology for planning and prioritization 14.1.1 Planning for a modeling and simulation project 14.1.2 Value systems for prioritization 14.2 Phenomena identification and ranking table (PIRT) 14.2.1 Steps in the PIRT process for modeling and simulation 14.2.1.1 Assembly of the team 14.2.1.2 Definition of the objectives of the PIRT process 14.2.1.3 Specification of environments and scenarios 14.2.1.4 Identification of plausible physical phenomena 14.2.1.5 Construction of the PIRT 14.3 Gap analysis process 14.3.1 Construct the gap analysis table 14.3.2 Documenting the PIRT and gap analysis processes 14.3.3 Updating the PIRT and gap analysis 14.4 Planning and prioritization with commercial codes 14.5 Example problem: aircraft fire spread during crash landing 14.6 References 15 Maturity assessment of modeling and simulation 15.1 Survey of maturity assessment procedures 15.2 Predictive capability maturity model 15.2.1 Structure of the PCMM 15.2.1.1 Representation and geometric fidelity 15.2.1.2 Physics and material model fidelity 15.2.1.3 Maturity assessment 15.2.2 Purpose and uses of the PCMM 15.2.3 Characteristics of PCMM elements 15.2.3.1 Representation and geometric fidelity 15.2.3.2 Physics and material model fidelity 15.2.3.3 Code verification 15.2.3.4 Solution verification 15.2.3.5 Model validation 15.2.3.6 Uncertainty quantification and sensitivity analysis 15.3 Additional uses of the PCMM 15.3.1 Requirements for modeling and simulation maturity 15.3.2 Aggregation of PCMM scores 15.3.3 Use of the PCMM in risk-informed decision making 15.4 References 16 Development and responsibilities for verification, validation and uncertainty quantification 16.1 Needed technical developments 16.2 Staff responsibilities 16.2.1 Software quality assurance and code verification 16.2.1.1 Who should conduct SQA and code verification? 16.2.1.2 Who should require SQA and code verification? 16.2.2 Solution verification 16.2.2.1 Who should conduct solution verification? 16.2.2.2 Who should require solution verification? 16.2.3 Validation 16.2.3.1 Who should conduct validation? 16.2.3.2 Who should require validation? 16.2.4 Nondeterministic predictions 16.2.4.1 Who should conduct nondeterministic predictions? 16.2.4.2 Who should require nondeterministic predictions? 16.3 Management actions and responsibilities 16.3.1 Implementation issues 16.3.2 Personnel training 16.3.3 Incorporation into business goals 16.3.3.1 Intrinsic information quality 16.3.3.2 Contextual information quality 16.3.3.3 Representational information quality 16.3.4 Organizational structures 16.4 Development of databases 16.4.1 Existing databases 16.4.2 Recent activities 16.4.3 Implementation issues of Databases 16.5 Development of standards 16.6 References Appendix: Programming practices Recommended programming practices Use strongly-typed programming languages Use safe programming language subsets Use static analyzers Use long, descriptive identifiers Write self-commenting code Use private data Use exception handling Use indentation for readability Use module procedures (Fortran only) Error-prone programming constructs Implicit type definitions Mixed-mode arithmetic Duplicate code Equality checks for floating point numbers Recursion Pointers Aliasing Inheritance GOTO statements Parallelism References Index "Advances in scientific computing have made modelling and simulation an important part of the decision-making process in engineering, science, and public policy. This book provides a comprehensive and systematic development of the basic concepts, principles, and procedures for verification and validation of models and simulations. The emphasis is placed on models that are described by partial differential and integral equations and the simulations that result from their numerical solution. The methods described can be applied to a wide range of technical fields, from the physical sciences, engineering and technology and industry, through to environmental regulations and safety, product and plant safety, financial investing, and governmental regulations. This book will be genuinely welcomed by researchers, practitioners, and decision makers in a broad range of fields, who seek to improve the credibility and reliability of simulation results. It will also be appropriate either for university courses or for independent study"-- Provided by publisher "Advances in scientific computing have made modelling and simulation an important part of the decision-making process in engineering, science, and public policy. This book provides a comprehensive and systematic development of the basic concepts, principles, and procedures for verification and validation of models and simulations. The emphasis is placed on models that are described by partial differential and integral equations and the simulations that result from their numerical solution. The methods described can be applied to a wide range of technical fields, from the physical sciences, engineering and technology and industry, through to environmental regulations and safety, product and plant safety, financial investing, and governmental regulations. This book will be genuinely welcomed by researchers, practitioners, and decision makers in a broad range of fields, who seek to improve the credibility and reliability of simulation results. It will also be appropriate either for university courses or for independent study"--Résumé de l'éditeur SUMMARY: The facts:-On 4 December 1971 a multinational European company, Klockner Industrie-Anlagen GmbH ("Klockner") and the United Republic of Cameroon ("Cameroon") signed a Protocol of Agreement under which Klockner undertook to supply, erect, and manage for a period of at least five years a fertilizer factory in Cameroon ready for use and with a production capacity of 157,000 tones per year.